Many natural phenomena are governed by forces on multiple spatial and temporal
scales. Yet, it is often not a computationally feasible option to describe the
intrinsically multiscale interactions via a deterministic model. Consequently,
there is a need to go beyond purely deterministic modeling and to use
stochastic processes to describe the unresolved scales of a system. Here the
considered process is assumed to be Markovian, i.e., the state probability
depends only on the previous state. Yet, the standard Markov model does not
allow to incorporate external influences that drive the considered system. A
modeling approach, addressing this issue, has been proposed by Illia Horenko
who suggested a model ansatz that incorporates available influences and is
applicable to identify time discrete Markov processes with a finite state
space. The unknown model matrices can be identified by means of an available
time-series via parametrization tools such as the FEM-BV clustering approach.
As in most realistic applications not all relevant quantities are directly
accessible; a central challenge is that such approaches are confronted with
the problem of missing information from unresolved or unmeasured scales.
Unfortunately, standard data-based analysis techniques often lack the option
to take these missing factors into account, leading to biased and distorted
results when confronted with this particular problem. As recently demonstrated
by Illia Horenko in the context of modeling discrete processes, such
systematically missing or implicit information can be taken into account via a
non-stationary model. In this thesis, the existing Markov regression framework
is extended for modeling of discrete stochastic processes with an additional
spatial component. In that context, the general problem of finding an adequate
data-based description of the considered spatio-temporal process in the
absence of relevant information is addressed. In purely time-dependent cases,
unresolved governing quantities lead to a non-stationary model structure. In
this thesis, it is shown that time as well as location-dependent processes
that are driven by unavailable influences can be adequately described via non-
stationary, non-homogenous models. A numerical approach to treat these new
structural properties of the model is proposed and implemented. Further, the
theoretically verified abilities of the proposed non-stationary, non-
homogenous Markov regression are also experimentally confirmed for an
artificial test system. In particular, the characteristic property to
recognize influences that are not directly accessible is experimentally
verified. Furthermore, the proposed framework is successfully used to gain a
deeper understanding of the dynamics underlying the arctic sea ice extent.